How To Write Your Own Equation in Algebra
By Kathleen Cantor, 03 Jul 2020
An equation is like a sentence, just using a different language. Mathematical equations are telling you to do something, giving you information on how to respond, or come up with an answer. It's up to you to learn this language. However, once you know it, you can write in this language! You, too, can write algebraic equations.
Parts of a Mathematical Equation
A mathematical equation, also known as a statement, shows that two separate expressions are equal. The two expressions are usually connected by the "equal-to" (=) sign. You can find the expressions in a mathematical statement by looking at the left-hand side and the right-hand side of the equation, using the = sign as the point of reference.
Example: 2x + 3 = 11
Can you identify the expression(s)? Can you locate the equal-to sign?
Balancing the Scale
It can be helpful to view equations as a scale in which weights are used to keep it balanced. The scale is balanced when what is placed on the right-hand side is equal in weight to what is on the left-hand side. Solving an equation can also be thought of in terms of adding weight to the left or right-hand side of the scale to balance it out.
Finding the solution to the equation involves manipulating the weights. Whatever is done on the right-hand side of the scale must also be done on the left-hand side to keep the scale balanced.
Your tools for this happen to be, most commonly, addition, subtraction, multiplication, or division.
The most popular equations are algebraic equations, in which letters are used as variables. Variables are placeholders that can assume any value. They can also be referred to as unknowns. You must solve for the correct value of the variable to make the equation true.
Let's look at an example.
4x + 2x = 12
The letter x is the variable. You must find the value of x. Finding this value will make the right-hand side of the equation equal to the left-hand side. Let's solve this together.
In this case, since both variables are the same, you can add the values on the left-hand side.
6x = 12
Now, we divide both sides by the value of '6', which will help us solve for x. Consider this to be a way to unravel the equation. Also, remember that whatever is done to one side must be done to the other.
x = 2
In this case, the value of x happens to be 2.
Go back into the original equation and substitute 2 in place of x.
(4 x 2) + (2 x 2) = 12
8 + 4 = 12
12 = 12.
We have successfully balanced the scale!
How to Write Equations in Algebra
The beauty of mathematics is that it provides a logical and sequential means of solving problems. Sometimes, the word problems that we encounter can be solved if they are represented in mathematical notation. A great way to do this by writing the problems as algebraic equations. Once the problem is represented as an algebraic equation, it can then be solved.
The first key lesson is to use letters to represent the unknown quantities of the problem.
1. Example 1
Jack left his house in the morning with $12. On his way to school, he bought ice cream. He had $8 remaining when he reached school. How much did he pay for the ice cream?
The amount Jack paid for the ice cream is unknown. This amount will be represented as x. We know that if we subtract the amount he paid for the ice cream from the money he left home with, we will have the amount of money he reached school with.
Step 1: Write the equation based on the relationships in the problem.
12 – x = 8
We can then find the solution to the equation by solving for x.
Step 2: Collect like terms by taking x to the right-hand side and bringing 8 to the left hand.
12 – 8 = x
Note that when a variable crosses the equality sign, the sign of the variable or number changes. Positive 8 turns into negative 8.
Step 3: Solve for x.
Follow operational rules to solve for the variable.
x = 4.
Jack paid $4 for the ice cream.
2. Example 2
Twice a number is 30.
This equation is written as:
2x = 30.
We know that a number multiplied by two ("twice") equals 30. The unknown number is x.
3. Example 3
Bob’s age 5 years ago was half the age he will be in 8 years. How old is Bob now?
This one is a bit trickier—the important thing when writing equations from word problems is to take it step-by-step. What we want to know is Bob’s age. It will be represented by x.
Bob’s age 5 years ago was x – 5.
Bob’s age 8 years from now is x + 8.
The problem states that his age 5 years ago was half the age he will be in 8 years.
x – 5 = ½(x + 8)
The equation also means “his age 5 years ago was half the age he will be in 8 years.”
The key to forming equations from word problems is by identifying the relationships between the various quantities in the problem. Your equation then brings together the relationships into a single math representation.
The key to writing your own equation in algebra is learning how to solve them. Like learning any language, this means you have to practice consistently. Try turning word problems into equations first. Then go ahead and make your own equation from scratch to solve. You may find that it ends up being great fun.
See the 2 Comments below.
24 Dec 2020 at 5:42 am [Comment permalink]
Multiplying through my 2
Group of terms
Dividing through by 3
11 May 2021 at 6:42 pm [Comment permalink]
When you grouped, you added x instead of subtracting x. When it changes side, it changes its positive / negative.
It should be 2x-1x = 18.
So x = 18